Counting supersolvable and solvable group orders

ABSTRACT

For more than 100 years, group and number theorists have been interested in quesitons such as: (a) If a group G has order $|\mathit{G}|=\prod p_i^{{\alpha }_i}$ (p_i distinct primes), what conditions on the primes p_i and their exponents α_i ensure that G is cyclic, or G is abelian, or G is nilpotent, or supersolvable, or solvable? (b) How fast does $\mathit{g}(\mathit{n})=|\{\mathit{m}\le \mathit{n}\mid$ every group G of order m has one of these properties$\}|$ grow as a function of n? Questions (a) and (b) have been answered when the property is either cyclic, abelian, or nilpotent. However, when the property is supersolvable or solvable, only question (a) has been fully answered. We greatly increase the current lower bounds for g(n) when the property is supersolvable or solvable. In the latter case, our lower bound is just below the best upper bound known. We used the MANA high performance computing cluster of the University of Hawaii at Manoa to greatly increase the current lower bounds for g(n) when the property is supersolvable or solvable.

KEYWORDS:

• Group theory
• supersolvable
• high performance computing

MATHEMATICS SUBJECT CLASSIFICATION:

• 20F16

Disclosure statement

No potential conflict of interest was reported by the author(s).